Threading formulas and definitions

This section covers threading formulas and definitions for how to calculate cutting speed, feed rate or any other parameter for your thread turning, thread milling or tapping operations. You also find the designation for different thread forms according to international standards.

Infeed

By dividing the full cutting depth of the thread into smaller cuts, the sensitive nose radius of the insert is not overloaded.

Example: By taking 0.23–0.10 mm (0.009–0.004 inch) in cutting depth per pass (radial infeed), the full cutting depth (a_p), and the profile depth of the thread (0.94 mm (0.037 inch)), is turned on a 1.5 mm (0.06 inch) pitch metric thread in 6 passes (nap).

Δ 𝑎 𝑝 𝑥 1 = 0.94 \sqrt 5 \cdot \sqrt 0.3 = 0.23

1st pass, infeed
0.23 mm

\[ \Delta a_{px1} = \frac{0.037}{\sqrt{5}} \cdot \sqrt{0.3} = 0.009 \]

= 0.009 inch

\[ \Delta a_{px2} = \frac{0.94}{\sqrt{5}} \cdot \sqrt{1} = 0.42 \]

2nd pass, infeed
0.42–0.23 = 0.19 mm

\[ \Delta a_{px2} = \frac{0.037}{\sqrt{5}} \cdot \sqrt{1} = 0.017 \]

0.017–0.009 = 0.008 inch

\[ \Delta a_{px3} = \frac{0.94}{\sqrt{5}} \cdot \sqrt{2} = 0.59 \]

3rd pass, infeed
0.59–0.42 = 0.17 mm

\[ \Delta a_{px3} = \frac{0.037}{\sqrt{5}} \cdot \sqrt{2} = 0.023 \]

0.023–0.017 = 0.006 inch

\[ \Delta a_{px4} = \frac{0.94}{\sqrt{5}} \cdot \sqrt{3} = 0.73 \]

4th pass, infeed
0.73–0.59 = 0.14 mm

\[ \Delta a_{px4} = \frac{0.037}{\sqrt{5}} \cdot \sqrt{3} = 0.029 \]

0.029–0.023 = 0.006 inch

\[ \Delta a_{px5} = \frac{0.94}{\sqrt{5}} \cdot \sqrt{4} = 0.84 \]

5th pass, infeed
0.84–0.73 = 0.11 mm

\[ \Delta a_{px5} = \frac{0.037}{\sqrt{5}} \cdot \sqrt{4} = 0.033 \]

0.033–0.029 = 0.004 inch

\[ \Delta a_{px6} = \frac{0.94}{\sqrt{5}} \cdot \sqrt{5} = 0.94 \]

6th pass, infeed
0.94–0.84 = 0.10 mm

\[ \Delta a_{px6} = \frac{0.037}{\sqrt{5}} \cdot \sqrt{5} = 0.037 \]

0.037-0.033 = 0.004 inch

\[ \Delta a_{px} = \frac{a_p}{\sqrt{n_{ap} - 1}} \cdot \sqrt{Y} \]

The infeed can be calculated according to the formula:

Δa_p = Radial infeed, cutting depth per pass

X = Actual pass (in a series from 1 to nap)

a_p = Total depth of thread + extra stock

nap = Number of passes

Y = 1st pass = 0.3

2nd pass = 1

3rd and higher = x-1

Pitch 1.5 mm (0.06 inch)
a_p = 0.94 mm (0.037 inch)
nap = 6

γ₁ = 0.3
γ₂ =1
γ_n = x-1

Symbol	Designation/definition	Metric	imperial
a_p	Infeed, full cutting depth	mm	inch
n	Spindle speed	rpm	rpm
V_c	Cutting speed	m/min
nap	Number of passes

Root/bottom
The bottom surface joining the two adjacent flanks of the thread
Flank/side
The side of a thread surface connecting the crest and the root
Crest/top
The top surface joining the two sides, or flanks.

P = Pitch, mm or threads per inch (t.p.i.)

The distance from a point on the screw thread to a corresponding point on the next thread measured parallel to the axis.

β = The thread profile angle

The angle between the flanks of the thread measured in an axial plane.

φ = Lead angle (helix angle) of the thread

The angle of the thread at a pitch diameter with a plane perpendicular to the axis.

Diameter parameters

d = Major diameter, external

D = Major diameter, internal

d₁ = Minor diameter, external

D₁ = Minor diameter, internal

d₂= Pitch diameter, external

D₂= Pitch diameter, internal

The effective diameter of the screw thread; approximately half way between the major- and minor diameters.

Lead angle (helix angle)

The lead angle, φ, (helix angle) is dependent on and related to the diameter and the pitch of the thread. This measurement can be represented by a triangle being unwound from the component. The lead angle is calculated according to the formula below.

\[ \varphi = \arctan\!\left(\frac{\text{number of starts} \times P}{d_2 \times \pi}\right) \]

Formula for calculating the lead angle.
N = number of starts

The same pitch on different diameters gives a different lead angle.

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Thread milling formulas and definitions

Cutting speed (v_c)

Metric (m/min)

\[ v_c = \frac{D_{\text{cap}} \cdot \pi \cdot n}{1000} \]

Imperial (ft/min)

\[ v_c = \frac{D_{\text{cap}} \cdot \pi \cdot n}{12} \]

Formulas for internal thread milling

Calculated version

Peripheral feed (mm/min) (inch/min)

\[ v_{fm} = n \cdot f_z \cdot z_c \]

Radial depth of cut (mm)(inch)

\[ a_{e\,\text{eff}} = \frac{D_m^2 - D_v^2}{4\,(D_m - D_{\text{cap}})} \]

Tool center feed (mm/min) (inch/min)

\[ v_f = \frac{v_{fm} \cdot (D_m - D_{\text{cap}})}{D_m} \]

Feed per tooth (mm)(inch)

\[ f_z = \frac{h_{ex}}{\sqrt{1 - \cos^2 \beta}} = \frac{h_{ex}}{\sin \beta} \]

\[ \beta = \arccos\!\left(1 - \frac{2 \cdot a_{e\,\text{eff}}}{D_{\text{cap}}}\right) \]

\[ D_{vf} = D_m - D_c \] \[ D_{vf1} = \frac{D_{vf}}{2} \] \[ D_{\text{cap}} \; \text{— cutter diameter} \] \[ D_m \; \text{— thread diameter} \] \[ D_w \; \text{— workpiece diameter} \] \[ a_{e,\text{eff}} \; \text{— effective radial depth of cut} \] \[ \beta \; \text{— engagement angle} \] \[ v_f \; \text{— tool center feed} \] \[ v_{fm} \; \text{— peripheral feed} \]

Thread milling with a roll into cut tool path, D_vf1

Formulas for external thread milling

Calculated version

Peripheral feed(mm/min)(inch/min)

\[ v_{fm} = n \cdot f_z \cdot z_c \]

Feed per tooth (mm) (inch)

\[ f_z = \frac{h_{ex}}{\sin \beta} \]

Engagement angle

\[ \beta = \arccos\!\left(1 - \frac{2 \cdot a_{e,\text{eff}}}{D_{\text{cap}}}\right) \]

Tool center feed(mm/min)(inch/min)

\[ v_f = v_{fm} \cdot \frac{D_m - D_{\text{cap}}}{D_m} \]

Effective radial depth of cut

\[ a_{e,\text{eff}} = \frac{D_m^{2} - D_w^{2}}{4\,(D_m - D_{\text{cap}})} \]

Tapping formulas and definitions

Metric

Spindle speed (n) (rpm)

\[ n = \frac{v_c \cdot 1000}{\pi \cdot D} \]

Feed rate

\[ v_f = P \cdot n \]

Inch

Spindle speed (n) (rpm)

\[ n = \frac{v_c \cdot 12}{\pi \cdot D} \]

Feed rate

\[ v_f = P \cdot n \]

Torque (M_d) (Nm)

\[ M_d = \frac{P^{2} \cdot D \cdot k_c}{8000} \]

Power (P) (kW)

\[ P = \frac{M_d \cdot 2 \cdot \pi \cdot n}{60} \]

Parameter	Meaning	Metric unit	Inch unit
M_d	Torque	Nm	in.lb
p	Pitch	mm	inch
D / DC	Nominal diameter	mm	inch
k_c	Specific cutting force	N/mm²	lbs/in2
n	Spindle speed	rpm (rev/min)	rpm (rev/min)
P	Power	kW	hp
v_c	Cutting speed	m/min	Feet/min
v_f	Feed rate	mm/min	in/min

Formulas for hole size

A basic hole size calculation can be made with the following formulas.

Basic hole size calculation for Cutting tap

\[ TD - TP = D \]

For example: To determine drill size for a cutting tap M10×1.5
10 − 1.5 = 8.5

\[ TD - \frac{0.975}{\text{TPI}} = D \]

For example: To determine drill size for a 1/4‑20 cutting tap

\[ 0.250 - \frac{0.975}{20} = 0.2013 \]

Basic hole size calculation for Forming tap

\[ TD - \left(\frac{TP}{2}\right) = D \]

For example: To determine drill size for a thread forming tap M10x1.5

\[ 10 - \left(\frac{1.5}{2}\right) = 9.25 \]

\[ TD - \frac{0.51}{\text{TPI}} = D \]

For example: To determine drill size for a 1/4-20 thread forming tap

\[ 0.250 - \frac{0.51}{20} = 0.2245 \]

Basic hole size calculation with thread height for Cutting tap

\[ TD - \frac{TH \cdot TP}{76.98} = D \]

For example: To determine drill size for a cutting tap M8x1.25 at 68% of thread height:

\[ 8 - \frac{68 \cdot 1.5}{76.98} = 6.896 \]

\[ TD - \frac{0.0130 \cdot TH}{\text{TPI}} = D \]

For example: To determine drill size for a 1/4-20 cutting tap at 65% of thread height.

\[ 0.250 - \frac{0.0130 \cdot 65}{20} = 0.2078 \]

Basic hole size calculation with thread height for forming tap

\[ TD - \frac{TH \cdot (TP / 2)}{147.06} = D \]

For example: To determine drill size for a cutting tap M8x1.25 at 68% of thread height:

\[ 8 - \frac{68 \cdot (1.5 / 2)}{147.06} = 7.422 \]

\[ TD - \frac{0.0068 \cdot TH}{\text{TPI}} = D \]

For example: To determine drill size for a 1/4-20 cutting tap at 65% of thread height.

\[ 0.250 - \frac{0.0068 \cdot 65}{20} = 0.2279 \]

Parameter	Meaning	Metric unit	Inch unit
D	Hole diameter	mm	inch
TD	Nominal thread diameter	mm	inch
TP	Thread pitch	mm
TPI	Thread pitch inch		inch
TH	Thread height	%	%

How to calculate thread height in percentage, %

Thread height in %, is the ratio between actual height and maximum height of the internal thread. The example below applies to ISO & UTS standard – 60° threads.

Example, M8x1,25

Maximum thread height according to standard is 6/8 H.

H = 0.866 x P

(H = Basic triangle height)

(P = thread pitch)

Maximum thread height is:

6/8 * (0.866 x 1, 25) = 0.811 mm

Actual thread height at a core hole of Ø6.9 mm:

(8 - 6,9) / 2 = 0.55 mm

Thread height is then (0.55 / 0.81) x 100 = 68%

Actual thread height

Rec core hole Ø

60° thread

Maximum thread height on internal thread (6/8 H)

Min core hole Ø according to standard

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Thread designations according to international standards

International standards

To ensure that the two (internal and external) halves of a threaded joint fit together properly to produce a connection capable of bearing a specified load, threads must maintain certain standards. International standards for thread forms have therefore been established for all common thread types.

ISO metric thread designations

The complete thread designation is made up of values for the thread form and the tolerance. The tolerance is indicated by a number for the tolerance grade, and letters for the tolerance position.

Examples

M16 - 6h

M16: Thread designation and nominal dimension

6h: Tolerance class for pitch and crest position

M10 x 1.25 5g6g

M10: Pitch

1.25: Tolerance class for pitch diameter

5g6g: Tolerance class for crest diameter

A fit between threaded parts is indicated by the internal thread tolerance class followed by the external thread tolerance class, separated by an oblique stroke.

Tolerance positions

The tolerance position identifies the fundamental deviation and is indicated with an upper-case letter for internal threads and a lower case letter for external threads. A combination of tolerance grade and position give the tolerance class. The values of the tolerance classes are given in the standards for the different threading systems.

ISO inch threads (UNC, UNF, UNEF, UN)

The UN system has three tolerance classes, ranging from 1 (course) to 3 (fine). A typical UN thread is designated as follows:

¼” 20 20UNC - 2A
¼” – Major thread diameter
20 – Pitch value: threads per inch (t.p.i.)
UNC – Indicates a course pitch
2A – Indicates a medium tolerance

ISO – unified (UN):

Loose tolerance: 1A (external threads), 1B (internal threads)
Medium tolerance: 2A (external threads), 2B (internal threads)
Tight tolerance: 3A (external threads), 3B (internal threads)

Different types of UN thread

UNC thread diameter with course pitch

UNF thread diameter with fine pitch

UNEF thread diameter with extra-fine pitch

UN thread diameter with constant pitch

Whitworth threads (G, R, BSW, BSF, BSPF)

Whitworth screw threads are now obsolete, but Whitworth pipe threads are a recognized international standard. There are two tolerance classes for external-, and one tolerance class for internal Whitworth pipe threads.

Whitworth pipe threads: BSW, BSF and BSP.F

Tolerance position
Fine: A (external threads), internal threads one class only
Coarse: B (internal threads), internal threads one class only

Whitworth pipe thread designations

These threads are divided into 2 groups:

Pressure-tight joints not made on the thread, ISO 228/1

Pressure-tight joints made on the thread, ISO 7/1

Threading formulas and definitions

Infeed

Diameter parameters

Lead angle (helix angle)

Thread milling formulas and definitions

Cutting speed (v_c)

Formulas for internal thread milling

Calculated version

Formulas for external thread milling

Calculated version

Tapping formulas and definitions

Formulas for hole size

Basic hole size calculation for Cutting tap

Basic hole size calculation for Forming tap

Basic hole size calculation with thread height for Cutting tap

Basic hole size calculation with thread height for forming tap

How to calculate thread height in percentage, %

Thread designations according to international standards

International standards

ISO metric thread designations

Examples

Tolerance positions

ISO inch threads (UNC, UNF, UNEF, UN)

ISO – unified (UN):

Different types of UN thread

Whitworth threads (G, R, BSW, BSF, BSPF)

Whitworth pipe threads: BSW, BSF and BSP.F

Whitworth pipe thread designations

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UNC	thread diameter with course pitch
UNF	thread diameter with fine pitch
UNEF	thread diameter with extra-fine pitch
UN	thread diameter with constant pitch

Threading formulas and definitions

Infeed

Diameter parameters

Lead angle (helix angle)

Thread milling formulas and definitions

Cutting speed (vc)

Formulas for internal thread milling

Calculated version

Formulas for external thread milling

Calculated version

Tapping formulas and definitions

Formulas for hole size

Basic hole size calculation for Cutting tap

Basic hole size calculation for Forming tap

Basic hole size calculation with thread height for Cutting tap

Basic hole size calculation with thread height for forming tap

How to calculate thread height in percentage, %

Thread designations according to international standards

International standards

ISO metric thread designations

Examples

Tolerance positions

ISO inch threads (UNC, UNF, UNEF, UN)

ISO – unified (UN):

Different types of UN thread

Whitworth threads (G, R, BSW, BSF, BSPF)

Whitworth pipe threads: BSW, BSF and BSP.F

Whitworth pipe thread designations

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Cutting speed (v_c)